Introduction

I come from a family with a fair number of gamers. Board games, role playing games, video games: someone in my family plays just every kind of game. A few years ago, I took a course on machine learning, where we spent a while reviewing all the probability rules that get covered in a Statistics 101 class, and while pondering the types of questions that DON’T get put into the 101 curriculum. This is about one of them.

Other than a coin flip, the most common probability example in introductory classes is based on a fair, uniform 6-sided die, where we assume that all sides are uniformly equal in probability.  For example:

The probability of rolling a “n” is 1/6, or 16.7%, for any “n” in [1, 2, 3, 4, 5, 6]. (1)

In these introductory classes, this uniform assumption is extended to example after example.

The probability of rolling “6” every time on k dice rolls is (1/6)^k. (2)

In order to roll a “4” in two die rolls, the first and second dice must be 1 and 3, or 2 and 2, or 3 and 1.  These are the only 3 combinations that produce a “4.”  Therefore, the probability of rolling a 4 on two dice rolls is the sum of the probabilities of rolling 1+3, 2+2, and 3+1. 

There are 36 combinations of rolling “a”+”b” on two rolls, and each are equally probable.

Therefore, the probability of rolling “4” is 3 out of 36 = 8.3%. (3)

Probabilities for a skewed die

The fundamentals of probability still hold if the assumption of uniformity is violated.  Skewed dice were designed (by the author) to illustrate how pervasive the uniformity assumption is.  These dice have been rolled a few hundred times to determine experimentally the probabilities of each number coming up. 

Figure 1: skewed dice, as saved on Thingiverse. Thanks to Josh Silverstein for saving, printing, sanding, and painting a set of model dice for me.

For Table 1 below, these probabilities have been rounded to a convenient number that is within the confidence interval of estimates of the true probability.  For example: the best estimate of the probability of rolling “2” is 19.67%, but I’m reporting it as 1 in 5, or 20%.  These rounded-off probabilities total 101%.  Deal with it.  More precise estimates are included in Appendix 1.

NOTE: for this paper, it is assumed that the numbers on an n-sided die are 1, 2, 3, …, n. For mathematicians and computer scientists who like to start counting at zero, you have my apologies.

Table 1:

Digit	Odds	Probability
1	1 in 3	33%
2	1 in 5	20%
3	1 in 10	10%
4	1 in 5	20%
5	1 in 9	11%
6	1 in 15	7%

For Equation (2), calculating the probability works the same way: the probability of rolling “1” three times on three rolls is

(1/3)³, or 3.7% (2a)

while the probability of rolling “6” three times is much less likely

(1/15)³, or 0.03% (2b)

Probabilities for multiple rolls

The probability of rolling a 4 on two dice rolls is STILL the sum of the probabilities of rolling 1+3, 2+2, and 3+1, but if the uniformity assumption is not valid, these probabilities have to be added up separately.  For example:

Let Pr(j, k) be the probability of rolling “j” as the sum of “k” die rolls.  The probability of rolling a 3 on one roll is therefore denoted Pr(3, 1), which from Table 1 is 10%.  The probability of rolling a 1 and a 3 is the joint probability of the two rolls:

Probability of 1 and 3 is the same as Probability of 3 and 1:

= Pr(1, 1) * Pr(3, 1) = (1/3)*(1/10) = .033 (3a)

Probability of rolling a pair of 2s on 2 rolls is the square of the probability of rolling a 2 on one roll:

= Pr(2, 1) * Pr(2, 1) = (1/5) * (1/5) = .04 (3b)

So the probability of rolling “4”:

= P( 1 and 3) + P( 2 and 2) + P( 3 and 1) = .033 + .04 + .033 = 0.106 —> 10.6% (3c)

This is further illustrated in Figure 2.  For any given number of rolls, the probability of rolling a certain total is an extension of the probability from each side.

Figure 2: Probability of rolling a total number j with k rolls of the skewed die. 

As a further extension of this, we see that there is a simple way to calculate the probabilities for an arbitrary k die rolls is iteratively.  If we have calculated all the probabilities for k-1 rolls, then:

(4)

A table of these probabilities is shown in Appendix 2.

Application of Central Limit Theorem

For a “fair” six-sided die, the average score over a large number of rolls is 3.5, which is the average of all sides: (1 + 2 + 3 + 4 + 5 + 6) / 6 = 3.5. For a skewed die, that formula is more properly expanded into a weighted average, by multiplying the value of each side by its probability.  Thus, for our 6-sided die,  The average of many rolls is:

Pr(1, 1) * 1 + Pr(2, 1) * 2 + Pr(3, 1) * 3 + Pr(4, 1) * 4 + Pr(5, 1) * 5 + Pr(6, 1) * 6 +

[ (.33 * 1) + (.2 * 2) + ( .1 * 3) + (.2 * 5) + (.11 * 5) + (.07 * 6)  = 2.8 (5)

So, for example, if you rolled the biased die 5 times, you can expect a total of around 2.8 * 5 = 14.

It goes without saying that to get a 14 on 5 rolls, this can happen by rolling 4 three times and 1 twice, or rolling 1 three times, 5 once, and 6 once, or any number of other ways.  However, as we roll the dice more times, it becomes more and more likely that there will be a combination of numbers which eventually approach the “true” average. For a more rigorous explanation of this, go somewhere else.

Another of the consequences of the Central Limit Theorem is that for a high enough number of rolls, the distribution of the sum of rolls looks more and more like a normal Gaussian distribution.  In Figure 2, this is clear as the k=4 rolls and k=5 rolls look more and more like a normal bell-shaped curve.

This brings up an amusing application.  In this case, the standard deviation is around 3.22, which is around 83% of the standard deviation for the fair die.  Also, the mean value from each roll of the die is around 79% of the mean for a fair die.  Within a margin of error, the sum of 6 rolls of this biased die will give around the same distribution of outcomes as 5 rolls of a fair die.

Calculations are left as an exercise for the reader (or see how it’s done here.)

The Jim’s Birthday problem

In Equation (4), we presented an iterative formula for Pr(j, k).  Theoretically, a closed form solution for calculating this should exist.  In honor of my brother’s 50th birthday, I’m calling this the Jim’s Birthday Problem.

APPENDIX 1: actual dice rolls for probability estimates

Based on 3 sets of 100 rolls.

APPENDIX 2: Pr(n, k) — probability of rolling “n” on “k” rolls

Given the single roll probabilities estimated in Appendix 1, the probabilities for rolling a given sum j on k rolls is calculated from Equation (4).

  Probabilities:

Everyone knows that inflation is “when prices go up” and everyone knows that inflation is the “natural” state of our economy. (More specifically, the Federal government maintains intentional feedback mechanisms to maintain a low positive inflation rate, but that’s an article for another day.) What fewer people have is a gut feeling for how much effect a small, cumulative inflation has on prices over the years.

In a recent Newsweek article about the proposed rise in the US Federal minimum wage, Senator John Thune is quoted as opposing the proposed $15 per hour level because as a kid, he made $6 an hour and did fine. As Newsweek points out, decades of inflation makes $6 in 1978 worth about the same as $24 in 2021,

This “candy was a nickel when I was a kid” misdirection is common, and it offends me. When people intentionally misuse math to make a false comparison between data from different decades, it bothers me even more that they do it to hide their own lack of empathy. Minimum wage was $2.65 in 1978. How many people have a feel for how much more that would be today?

I decided to put together a handy conversion table.

The US Bureau of Labor Statistics has more raw data than most of you are going to want for a “market basket” calculation they use to compare prices over time. They calculate several flavors of Consumer Price Index (CPI), and while there are some issues, it’s been a standard for comparison since shortly after WWII. I’ve picked series CUSR0000SA0 for this comparison: “All items in U.S. city average, all urban consumers, seasonally adjusted”.

The CPI was 262.231 in January 2021.

At some point in the past, it was half of that, at 131.1, so a dollar at that time was worth $2 now.

From the tables, that date was August 1990, so $1 in 1990 would be worth around $2 now. The rest of my table follows the same format: CPI in October 1951 was 26.2, one tenth of today’s, so $1 then would be worth $10 now.

Date	CPI	Multiplier
2021 January	262.231	1
2000 December	174.6	1.5x
1990 August	131.1	2x
1981 January	87.2	3x
1978 July	65.5	4x
1975 February	52.6	5x
1973 April	43.7	6x
1966 September	32.8	8x
1951 October	26.2	10x
1947 January	21.5	12x

Note: the 1970s were a time of consistently high inflation, and everyone was aware of it. If your wages weren’t going up 10% or more some years, you were making less than the year before. That shows up in the table: $1 in February 1975 is worth $5 today, while $1 in April 1973 is worth $6 today.

To make a long story short, if anyone tells you about wages and prices in the good old days when they were a kid, be prepared to multiply it 2x for folks who got their first job in 1990, triple it for first jobs around 1981, and multiply it by up to 10x for some of the oldest boomers.

In recent court filings about the Georgia election, Christos A. Makridis made some wild accusations about voter fraud which are rebutted in great detail by Professor Jonathan Rodden of Stanford. Amid his piece-by-piece analysis, Rodden addresses a claim about Benford’s Law.

Madrikis writes:

One diagnostic for detecting fraud involves Benford’s law. In the case of election fraud, that means looking at the distribution of digits across votes within a specified geography. Using precinct level data
for Georgia, my research identified 1,017 suspicious precincts out of 2,656 when we look at advance ballots.

In his response, Rodden can be seen almost visibly throwing up his hands:

Studies using the socalled First Digit Newcomb-Benford Law have come under heavy criticism for
electoral applications… The type of analysis undertaken by scholars in this literature cannot be written up in a breezy paragraph that lacks crucial details, such as Dr. Makridis’ brief exposition on page 4 of his report.

In this post, I will discuss some rules of thumb for when Benford’s Law will be a reasonable thing to expect, and why this isn’t one of them.

For those unfamiliar with this “law”, let’s start by noting that it’s not a causal law like “gravity points down” or a legislative law like “don’t drive drunk” but rather an observation that many data sets follow a particular interesting pattern. It’s not an enforced rule, but an observed one. Here’s what it looks like.

Suppose you gave everyone in the local high school 24 cookies, and told them to eat what they want and take the rest home. When they left the school, you tallied how many cookies each person had left. Chances are, many of the kids will eat no more than 4. Their remainder will be “twenty-something,” or some number starting with 2. Of the others, almost all will eat fewer than 14 cookies, leaving a number starting with a 1. There may be a few with zero left, but not many. There will likewise be few who eat 20 of the 24 and take home the rest: there will be few remainders of 4.

That’s basically all it is. For a lot of data sets, you can expect the first digit to be “usually” some particular digit or follow a certain pattern. If you are looking at a list of the tallest buildings in the world (in feet), you can expect the first digit to be almost always 1. There is a very short list of buildings over 2000 feet, and anything under 1000 just won’t make the list.

Anyway, Benford’s Law is the observation that for certain types of data, the first digit is 1 more often than any other, and 2 is second most common, etc. Madrikis is suspicious because the number of people voting at each precinct doesn’t follow that pattern. Rodden rolls his eyes. If it’s not yet clear why, let me finish.

Georgia voters are assigned to one of 2652 precincts. In the 2020 presidential election, just under 2.5 million of them voted for each presidential candidate. If evenly distributed, that would be around 940 registered voters per precinct. If you tally the number of votes per candidate at each precinct, you can expect a lot of those tallies to start with 9. A lot will also start with 8, and a pretty good number will start with 1. You would expect the number of precincts with totals starting with 2 will be minimal: it either means 2000+ voters went one way, or under 300. It happens, but in Benford’s Law data sets, 2 is the second most common first digit. That just isn’t going to be the case here.

That’s all — common sense. If someone wants to dig up the precinct data and confirm how it actually fell out, feel free. Please let me know. I think the GA Secretary of State has that. I just know I wouldn’t apply Benford’s Law here, and anyone using it as “proof” of fraud because election results didn’t turn out the way they wanted is ignorant of how it is applied, or is trying to bamboozle people, or both.

Folks, here’s another “I read it so you don’t have to”. For the TL;DR crowd, the claims that statistics show a one in a quadrillion chance of the election turning out the way it did comes from a painfully abused misapplication of math. It’s nonsense.

The Supreme Court filing by Texas against Pennsylvania, Georgia, Michigan, and Wisconsin has, among other things, the following statement by Charles J. Cicchetti, an economist:
_{“I was asked to analyze some of the validity and credibility of the 2020 presidential election in key battleground states…
I determine the Z-scores comparing the number of votes Clinton received in 2016 to the number of votes Biden received in 2020. The Z-score in 396.3. This value corresponds to a confidence that I can reject the hypothesis many times more than one in a quadrillion times that the two outcomes were similar.”}

Over the last 3 days since the filing, I’ve heard “quadrillion” more times than I usually do in a decade. It’s now being thrown around as more “evidence” that there was something fishy about the ballots, while taking the subsequent leap that the whole states’ elections should be thrown out. That’s not what the numbers say.

Here’s what Dr. Cicchetti’s analysis actually did. It demonstrated that the outcome of the 2020 election cannot be assumed to have come from identical conditions as the 2016 election. That is all. In case any of you would not have already concluded that, here is a little more about what he did.

Background, first of all. The Z-score test that was applied is appropriate for analyzing random events, which is not necessarily the right tool for the job. Here’s how the tool is supposed to be used.
Suppose you have a slightly damaged coin and you want to know if it flips 50:50 fair. You flip it 100 times, counting the number of heads, and then do another 100 flips, counting again. Common sense says that with an unbiased coin, you “should” get around 50 heads from each series, but we also know that it’s in the nature of random coin flips not to get exactly 50 every time. What if you got 54 heads followed by 47 heads? Weird? Probably not. Most people would say that’s just what happens sometimes. There’s no reason to think that there’s anything funny going on with results like that. But what if you got 92 heads followed by 94 heads? Clearly there’s something fishy about the coin. 90 heads followed by 90 tails? Most people would agree that not only is that a heavily tricked-out coin, but you can’t be using the same trick coin for both runs. That’s where statistical tests come in. The Z-test is one of the tools used to determine whether it’s safe to conclude that a (possibly biased) coin tossed in two sequences are likely to have acted differently.

(I think my readers will forgive me if I leave out the calculation of the probabilities above. Suffice it to say that anyone with a college class in statistics has the background to look up how to do it. For what it’s worth, the chances of getting EXACTLY 50 heads on 100 throws of a fair coin is only about 8%, and getting 46-53 out of 100 is only around 43%, so the 54 & 47 example is totally not fishy. Getting further and further from 50:50, we find that 88 or more heads is where we break the one in a quadrillion threshold, while 92 or more out of 100 is around a one in a 6.2 quintillion. 92/100 followed by 94/100 is what I’d call “once in never.”)

Suppose we accept the filing’s statement that in 2016, Clinton received 1,877,963 votes, which is 45.9% of the total cast. Suppose we imagine that these votes came about from a long series of coin flips from a slighty bent coin thrown over 3 million times and which come of “not Trump” 45.9% of the time. Here is what Cicchetti tested: If we know in advance that this exact same biased coin was used again, what is the probability that when we throw it 4 million times in 2020, it comes up with “not Trump” 49.5% of the time instead of close to 45.9%?

The answer will not surprise you. The chances are one in … you guessed it … one in LOTS. Lots and lots and lots. Any statistician would say there is no reasonable possibility that this was the same coin. One in a quadrillion is actually optimistic by more than a billion times a billion times a billion. It’s “yuge.” With real coin throws, more throws you have, the more closely the average matches the true average of the coin. So with 4 million throws, while the true probability of throwing “not Trump” is 45.9%, the probability of being outside of 45.9% +/- 0.2% is about 1 in 16,000, and the probability of being outside of +/-0.4% is already in the neighborhood of 1 in a quadrillion. For the average to move by 3.6%, the only reasonable conclusion is that the assumptions of this mathematical model don’t match reality.

So that’s what we conclude. We conclude that conditions have changed. We conclude that the voters felt that the Clinton-vs-Trump election was not identical to the Biden-vs-Trump one. Some people changed their minds. Different people thought it was important enough to vote. In any case, the conclusion is clear. The 2020 election was not the 2016 election. His assumptions don’t match reality.

That’s really all that Cicchetti can say. But there’s more.

Elections, you all understand, are not coin flips. They are not random. They are the fully deterministic outcome of a single set of choices made by a single set of voters, each getting one vote, and at the end of the day, those choices are counted. There are no statistics to tabulate. They are what they are. If you want to say that a re-vote by the same electorate on a different day under different conditions would have different results, that’s just wind. Those hypothetical elections don’t exist, and don’t matter. Dewey could have beaten Truman.

Polls leading up to elections get analyzed statistically. If you want to predict the winner, you have to assume that limited number of people you polled are a true reflection of the millions who will actually vote, and modeling those polls like a series of coin flips is one of the tools in the tool box. If I call 100 people on their land lines, and ask them who they voted for for president 4 years ago, and all 100 say “Mickey Mouse”, I may call the election as a sure thing. If that’s not who wins the election, then the assumptions I made were wrong.

That is all.

47 years! Forty-seven years! 47 yeeeeeeears!!

The latest keep-on-stabbing point against Biden is that he’s taken a government salary for 47 years, and since he’s done nothing for the country in that time, he’s “taken” more than Trump has through any tax finagling he has managed to perform. This is an argument for people bad at math.

It’s also an argument for people who FEEEL that Biden is a bad guy for spending a lifetime of government service, learning patience and statecraft. I know that a reasoned rebuttal isn’t going to change their feelings. Still, bad math bothers me, so I have to take a moment for quick riposte.

We can leave aside for a moment the fact that Biden’s constituents re-elected him again and again, and that THEY thought he was doing something for them. For that rebuttal, feel free to read any of the first 5 articles you get if you Google “Joe Biden’s Accomplishments.” Let’s just stick to arithmetic.

Trump claims to be a billionaire, making hundreds of millions of dollars a year. By the most kind, conservative interpretation, if he paid 10% on $100 million a year, he’d pay $5 million a year. Almost all of us pay a higher rate than that, and he claims to make more than that, but let’s just leave that number there for comparison. By this rule of thumb, he’s found loopholes to avoid virtually all of that $5 million a year for most of the last 20 years’ taxes.

In comparison, the cumulative total of Biden’s earnings, earned by a full-time job through multiple re-elections over those 47 years is less than the last 2 years’ tax loopholes.

1. Biden’s salary for 8 years as Vice President was 8 x $230,700 = 1.85 million.

2. Suppose that at maximum the other 39 of those 47 years were paid at today’s senator rate (they weren’t) then 39 x $174k = 6.79 million.

3. For those who need me to do the rest of the arithmetic for them, thats 1.85 + 6.79 = 8.64 million cumulative total salary.

So what we’re hearing from the 47-year crowd is that if Trump cheated the government out of at least $15 million since becoming president, that’s ok because the other guy got over half of that for 47 years of salary. That’s a fine argument if you’re bad at math. That’s a fine, fine argument if you FEEEEL that only Trump can save you. Just remember that we recognize that you’re comfortable saying irrational, emotional, illogical things and that the rest of the world looks down on you.

It’s been 3 months since I started my last Covid post on Facebook with this:
“You know what sucks the most about being in tune with data? Seeing the future.”

For those of you not looking, or not seeing trends, here’s more of the same. For the last month or so, there’s been a band of outbreaks from Montana through Arkansas which we can safely call the the opening month of the third wave.

There are literally 29 counties in the USA where over 1% of their population was diagnosed with covid THIS WEEK. 9 counties in North Dakota, 7 in Montana, 5 in South Dakota, 4 in Nebraska, 2 in Kansas, and 2 in Wisconsin. In Toole County, Montana, 2.5% of the county got it this week.

In comparison, my home town of Seattle was the first place in the USA to have over a dozen confirmed cases in February, and our county has been holding steady at around 0.05% a week, just having passed 1% cumulative cases. Over 80% are from months ago, and have recovered and gone home.

Anyone who knows anything about the spread of disease knows that these mostly-rural counties have a safety advantage because of cheap land and open spaces. Montana has no 50-story apartment buildings or single acres where a quarter million people are breathing each others’ air every day. If they chose to be as careful as New York, where 0.05% of the state got sick this week, they could do that. Montana is at least as rural as New Hampshire, where state-wide infection rate is at 0.7% cumulative, and their worst county this week was at 0.05%.

Europe is not immune, and it may be about to get worse there, too. Belgium and the Czech Republic had infection rates this week worse than any of our states besides North and South Dakota. Ignoring a couple of tiny countries, Netherlands and France are the 3rd and 4th most noteworthy this week, with average infection rates as bad as our 6th-to-10th worst states.
Taken as a whole, they’re in for a rough winter. As are we all.

There are those who say they aren’t worried about the spread because the death counts haven’t been rising with the illness rate. There’s actually 3 reasons why this is misinformed optimism.

The first reason that death rates haven’t risen in the last few weeks is that a tremendous number of new cases are among young people. Median age of those diagnosed dropped from 46 at the beginning of May to 36 at the end of August, and continues to go down.

Younger victims don’t die as often. They’re the Typhoid Marys and the plague rats, any many of them believe themselves to be invulnerable, but even the asymptomatic ones can bring it home and spread it to family and friends. It’s no coincidence that the infection rate started going up shortly after kids started going back to school.

The second reason that death rates haven’t risen much is timing. In each of the first 2 waves, death rates trailed behind infection rates by 2-3 weeks. So we’re seeing the beginning now. More is on the way.

The 3rd reason is the concerted effort by our current government not to call anything a covid death if there is any excuse not to. As reported here in August, and extended here more recently, the data is clear: we’re under-reporting on purpose.

As always, more data sources are out there: here for example, and here. As always, keep your distance, wear a mask, wash your hands. Don’t be a plague rat.